White smoke and bad shake

I would start with a compression test to at least know if and which cylinder it is first. Then perhaps the leak-down test instead of having to do the leak-down test on every cylinder cause that would take more time having to place every single piston at TDC compression. It could though ending up being the intake gasket leaking the coolant. Or just remove all the spark plugs and the one with the coolant on it, start there doing the compression test and then leak-down if need be.
Make sure when you are all done, you drain that motor oil and change that filter to get all that coolant out that leaked pass the piston rings cause if left down there is will eventually wash away the motor oil on the bearings and ruin them.

Does seem to have those symptoms. You can get exhaust analyzed for anti freeze trace. You could also pressure test system to see if a leak is present.

You are welcome and good luck!

They make a lot of liquid head gasket repair products these days. Yours being a mild case is in your favor for such a fix. Especially if it's a crack and not a gasket?

I do not recommend you drive it any further at all. Look into renting a tow bar, a tow rope if legal in your state or something else.
If you have to absolutely drive it, at least do an oil/filter change first to get all the coolant out of the oil pan, make sure the coolant is topped off, too and keep an eye on the temperature gauge, check engine light, etc, and any steam from under the hood, engine knocking or any sudden more white smoke out the tailpipe and drive as easy as you can.

White smoke is a sign of burning engine coolant, typically from a leaking head gasket. Has the level dropped in the coolant reservoir?

Have you had to add coolant? White smoke indicates there is coolant getting into the combustion chamber. My guess is a bad head gasket.

Let me know about coolant usage. Also, check to make sure the coolant isn't mixing with the oil.

Let me know what you find.

Joe

Yes they are separate it sound like you have a blown head gasket. Is the coolant low?

Please let me know, Ken

White smoke is generally related to coolant burning, are you losing any coolant?

Pinging can be from a multitude of reasons causing a lean condition. MAF contamination, excessive build up of carbon to name a couple. Thry an air induction service for the pinging, It's good maintenance regardless.
I would try what I call an air induction service.
I like to use castle hydroblast. You spray it into a vacuum line such as the brake booster line if nothing else is accessible. Full spray for 3 minutes, stall the engine with it or have someone kill it while your spraying. Let it set for a few hours then take for a short drive in low gear, revving the rpms to help break it up. First use Castle cleansrite to clean the throttle bore. Try to get the throttle plates scrubbed good with a brush, while your spraying. Use a rag to wipe the carbon out between brushings. Seafoam is another good product, but smokes a lot.

IF you can't find Castle, here is my rep's email:
robw@castlepackspower. Com

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I don't know where you live, but if it's cool outside, it could just be condensation from the exhaust system. If you have to add coolant often, it could still be a head gasket even though it isn't getting into the oil.

this guide will help

https://www.2carpros.com/articles/white-smoke-or-steam-coming-from-the-exhaust-pipe